OK I am in a similar situation, but not living together full time yet, but 75% of the time. We both work, we both have 2 kids, mine 100% his 50%. Kids ages DD10, DD9, DSD9, DSS6. We will all be moving to his house eventually, but most likely we will end up at mine for a while first.
The thing is that we both come from big families so we know how they work. The kids do everything. So, washing - someone goes and gets it all. I sort it for the machine -although they could. They sort and fold it, put the ironing pile together. We outsource the ironing and cleaning - 7 hours a week.
Cooking - I'm in charge, but they do help with everything. They lay and clear the table. Load and unload the dishwasher. They work as a team. No-one gets to walk away if there is stuff to be done.
They mow the easy bits of the lawn. We have some help in the garden. My DP does most of the gardening and an enormous amount of DIY - we have 3 holiday rentals, as well as both working full time.
We often have "helpers" or au pairs staying, and they then take on the cooking. However sometimes they are more work than they are help.
Like you we do have precious little time alone together. And if the kids were any older no doubt we'd get even less time together.
Really my best advice is to outsource what you can, and the kids get a load of tasks. No phone/tablet/computer time until all the tasks are done.